


Dynamical Systems and Applications / Systèmes dynamiques Org: Michael A. Radin (RIT)
 WILLIAM BASENER, Rochester Institute of Technology
Booming and Crashing Population Dynamics and Easter Island

We present a modified predatorprey ODE modeling an isolated
population and its resources. Numerical results using appropriate
parameters match archeological data for the human population on Easter
Island. This is in contrast to standard models, which do not predict
the booming and crashing of the Easter Island population. A complete
qualitative analysis of the model is presented, including a 2D
degenerate Hopf bifurcation and a Lyapanov function. Other
applications of this model are also considered.
 LORA BILLINGS, Montclair State University
Stochastic Epidemic Outbreaks: Why Epidemics Are Like Lasers

Many diseases, such as childhood diseases, dengue fever, and West Nile
virus, appear to oscillate randomly as a function of seasonal
environmental or social changes. Such oscillations appear to have a
chaotic bursting character, although it is still uncertain how much is
due to random fluctuations. Such bursting in the presence of noise is
also observed in driven lasers. In this talk, I will show how noise
can excite random outbreaks in simple models of seasonally driven
outbreaks, as well as lasers. The models for both population dynamics
will be shown to share the same class of underlying topology, which
plays a major role in the cause of observed stochastic bursting.
 STEFANELLA BOATTO, Dept. of Mathematics and Statistics, McMaster University,
1280 Main Street West, Hamilton, Ontario L8S 4K1
Periodic solutions of the Euler's equation on a sphere

Classes of steady and periodic solutions are investigated for the
incompressible Euler equation on a sphere. Of particular interest is
the stability of such solutions. The study makes use of an infinite
dimensional Hamiltonian formulation of the vorticity equation when the
rotation of a planet is taken into account.
 ERIK BOLLT, Clarkson University, Potsdam, NY 136995815
Mapping Transport Activity in Stochastic Dynamics, Directly
from The Transfer Operator and Learning Noise Induced
Multistability

Associated with a dynamical system, which evolves single initial
conditions, the FrobeniusPerron operator evolves ensemble densities
of initial conditions. We will present our new applications of this
global and statistical point of view: Wellknown models have been
found to exhibit new and interesting dynamics under the addition of
stochastic perturbations. Generalizing the FrobeniusPerron operator
to stochastic dynamical systems, we develop new tools designed to
predict the effects of noise and to pinpoint stochastic transport
regions in phase space in the absence of global manifold information.
 BERNARD BROOKS, RIT
Discrete Population Dynamic of Easter Island

The continuous population dynamic of Easter Island has been
investigated in order to try and explain the unusual historic
population trends noted on the island. Archaeological evidence
suggests that the island's population boomed and then, shortly before
the arrival of Europeans, crashed almost to zero. The continuous
model yielded very interesting results and demonstrated that there did
exist a tipping point where the inhabitants could harvest too much of
their resources resulting in the observed data.
This talk will focus on the analysis of the discrete version of that
model. The 2dimensional first order discrete dynamic contains the
familiar logistic map to model the human population. What is not so
familiar is that the resources that the population's carrying capacity
depends on will also be governed by logistic growth. This
modification to the wellstudied logistic growth concept provides a
much more realistic model of an isolated population feeding on
biological resources. In addition to that added realism, the dynamic
is mathematically rich hinting at the possibility of chaos.
 LEO BUTLER, Queen's University, Kingston, ON K7L 3N6
Invariant fibrations of geodesic flows

A flow f_{t} : M ® M is weakly integrable if there is
an open dense set L Ì M that is fibred by invariant tori. If
G = ML is a tamely embedded polyhedron and the invariant tori
are kdimensional, then f_{t} is ksemisimple.
Theorem: Let (S, g) be a compact riemannian 3manifold.
If its geodesic flow is 3semisimple, then p_{1}(S) is almost
2step polycyclic.
The proof uses an argument due to Kozlov and Taimanov. The
EvansMoser classification theorem and work by Bolsinov and Taimanov
implies a partial converse.
 HAMIDEH D. HAMEDANI, Department of Statistics, Faculty of Mathematical Sciences,
Shahid Beheshti University, Tehran, Iran
Stopped Semimonotone Nonlinear Stochastic Integral
Equations with Martingale Noise

The theory of stochastic differential equations in Hilbert and Banach
spaces have a wide range of applications in stochastic modeling to
describe the dynamic of random phenomena studied in science and
engineering. In this context, there are many initial value problems in
which time is stopped by a given stopping time. For instance, they
may occur for parabolic and hyperbolic type PDEs with a stopping
condition at the boundary of the region, or for stochastic systems in
bounded regions with absorption at the boundary. Hence an
existenceuniqueness result plays an important role in this direction.
Consider the following semimonotone nonlinear integral equation:
X_{t} = U(t,0) X_{0} + 
ó õ

t
0

U(t,s) F(s,X_{s}) ds + 
ó õ

(0,t]

U(t,s)G(s_{},X_{s}) dM_{s} + V_{t} t £ t, 
 (1) 
where
(a) U(t,s) is a contractiontype evolution operator
generated by A(.).
(b) M is a Kvalued cadlag square integrable martingale
with arbitrary jump sizes, G Î L^{2} (K; H;
P, M) where K and H are real separable Hilbert
spaces, and P is the salgebra of predictable
sets. F_{t}(.) = F(t,w,.) : H ® H is
demicontinuous, semimonotone and G_{t}(.) = G(t,w,.) : H ® L^{2} (K; H; P, M) is
Lipschitz. Suppose that a linear growth condition as well as
appropriate measurability assumptions hold for both F and G.
(c) V_{t} is an Hvalued cadlag adapted process and X_{0} is
a random variable.
Using Picard iterated method, we prove that the integral equation (1)
has a unique strong measurable solution. We also show the continuity
of the solution with respect to the initial condition.
 HAROLD HASTINGS, Department of Physics, 151 Hofstra University, Hempstead, NY
115491510
Microscopic fluctuations and pattern formation in a chemical
oscillator

The spontaneous formation of order in the form of spatial
concentration patterns in an unstirred chemical medium, supported by
dissipation of chemical free energy, has been considered often since
early work of Turing and Prigogine's group on nonequilibrium
thermodynamics. The bestknown experimental example is the
oscillatory BelousovZhabotinskii reaction, in which target patterns
of outwardmoving concentric rings are readily observed. Question:
can "microscopic" fluctuations nucleate targets, or is a catalytic,
nucleating heterogeneous center required? Vidal and Pagola observed
spontaneous activity with no visible nucleating particles; however
Zhang, Forster and Ross argued that fluctuations cannot nucleate
targets far from bifurcation points.
We demonstrate that microscopic fluctuations can nucleate targets by
simulating and analyzing dynamics as the system passes slowly through
a Hopf bifurcation.
Joint work with Richard J. Field (U. of Montana), Sabrina G. Sobel
(Hofstra U.). Partially supported by the NSF.
 VASSILIOS KOVANIS, Department of Mathematics and Statistics, Rochester Institute
of Technology
Coherence Collapse for Fun and Profit in Optical Systems

Semiconductor lasers have a wide range of applications in novel
optical communication devices, however tiny amounts optical feedback
is enough to destabilize their normal output and cause a phenomenon
called coherence collapse. By contrast to other commercial lasers
such as gas and solid state lasers, the dynamical mechanisms causing
these instabilities are less understood. Direct experimental
observation of the intensity of the diode laser output versus time is
a challenging issue but Fourier domain optical measurements combined
with numerical studies of simple nonlinear rate equations have shown
that these instabilities are typically the result of successive period
doubling bifurcations. We analyze these bifurcations using modern
asymptotic techniques by taking advantage of the natural parameters of
the diode laser equations. Implications of these theoretical findings
to modern novel photonic applications including bandwidth enhancement,
photonic analog to digital conversion and communication with chaotic
signals will be discussed.
 HERBERT KUNZE, University of Guelph, Guelph, ON
A collage coding approach to an inverse problem for systems
of firstorder linear PDEs

In fractal imaging, one approximates a target image by the fixed point
of a contractive map which seeks to represent the image as the union
of shrunken and distorted copies of subsets of itself. In the
literature, the process is named collage coding. In recent work, I
have used collage coding ideas to solve some inverse problems for
ordinary differential equations and integral equations. In this talk,
I will discuss some recent progress made with this approach for a
general inverse problem for systems of firstorder linear (and perhaps
quasilinear) PDEs.
 FRANKLIN MENDIVIL, Math. Dept., Acadia University, Wolfville, NS B4P 2R6
Convergence of a Stochastic approximation scheme

Given a finite graph G and a probability distribution p on the
vertices of G, we discuss an iterative scheme which generates a
dynamical system whose empirical occupation distribution
approaches p.
 TAMAS WIANDT, Rochester Institute of Technology
Conley Decomposition and Liapunov Functions for Closed
Relations

We present a short overview of the theory of dynamics of closed
relations on compact Hausdorff spaces. We establish generalizations
for some topological aspects of dynamical systems theory, including
recurrence, attractorrepeller structure, the Conley decomposition
theorem, Liapunov functions and intensity of attraction.

